Buggé: Electrostatics 4
What Determines the Strength of Electrostatic Interactions?
4.1 Find a Pattern​ In 1785 Charles Coulomb used​ ​a torsion balance
(shown to the right) to determine how the force that one charged sphere
exerts on another charged sphere depends on the magnitudes of the charges
of the two charged objects and their separation.
In Coulomb’s time scientists could did not know how to directly measure
electric charge, and the unit for charge did not exist. So instead, Coulomb
was able to determine relative charges by touching a charged metal sphere
with an identical uncharged metal sphere.
a. What is the relative charge on each sphere? Explain the mechanism
behind his method of dividing the charge. Why did he use metal
spheres rather than foam spheres?
b. The table below provides simplified data that resembles what Coulomb might have collected. ​Plot 3
graphs to find patterns in the data and devise a mathematical relationship based on these
observations.​ (Remember to decide which variables are independent and which variable is the
dependent. Then examine the dependent variable as you​ change only ​one independent variable at a
time.)
​
c. Write
​
an expression that shows
​ ​ ​ Fq1 on q2​ as a function of ​q1​, ​q2​, ​r, and
​ a proportionality constant, ​k.
d. This expression is used to determine the magnitude of the electric force. Should we include the signs
​ of the electric charges ​q1​ and ​q2​? How do we determine the direction of the electric force?
Buggé: Electrostatics 4
4.2 Reason (​The following activities were borrowed from E&M TIPERs)
The diagrams show two charged objects and their separation. Rank the force that the left object exerts on the
right object from the STRONGEST to the WEAKEST force. Explain how you made the ranking.
A. 1.
2.
3.
2.
3.
2.
3.
2.
3.
2.
3.
RANKINGS:
B. 1.
RANKINGS:
C. 1.
RANKINGS:
D. 1.
RANKINGS:
E. 1.
RANKINGS:
For each of the cases compare the force that the left object exerts on the right object to the force that the
right object exerts on the left object. Draw and label force diagrams. Explain how you know.
​
Buggé: Electrostatics 4
4.3 Reason
​
​In each diagram below are small charged objects fixed on a grid. All charged objects ​q are identical. All
other charged objects are a multiple of ​Q. Draw the electric force diagram for each charge ​q, then rank the magnitudes
of the net electrical forces exerted on ​q. When you are done, identify third law pair forces for each diagram.
Buggé: Electrostatics 4
Coulomb’s law: ​The magnitude of the electric force that object 1, with electric charge ​q1​, exerts on
object 2, with electric charge ​q2​, when they are separated by a center-to-center distance ​r is given by the
expression below. Note that this is also the equal magnitude electric force that object 2, with electric
charge ​q2​, exerts on object 1, with electric charge ​q1​:
where ​k = 9.0 x 10​9​ N•m​2​/ C​2​. We assume that the objects are much smaller than their separation ​r (i.e. point
particles).
μC = 10​-6​ C
nC = 10​-9​ C
pC = 10​-12​ C
4.4 Reason ​In each diagram below are small charged objects momentarily fixed on a grid. A charged object
of +​q is placed on the left a distance ​d from the +3​q charged object. The two objects are released
simultaneously in a frictionless environment. Next to each particle is its mass ​m (or some multiple of ​m).
a) Compare the magnitude of the force that particle A exerts on particle B with the force that particle B
exerts on particle A. Write the result of your comparison as a ratio.
b) Rank the magnitude of acceleration of each particle. Show all your work.
Buggé: Electrostatics 4
4.5 ​a) Compare the electric force exerted by the proton on the electron in a hydrogen atom to the to the
gravitational force. A proton has a charge + 1.6 x 10​-19​ C and mass 1.7 x 10​-27​ kg. An electron has a charge
– 1.6 x 10​-19​ C and mass 9.1 x 10​-31​ kg. Distance between Proton and Electron ~ 10​-10​ m.
b) What about the gravitational force exerted by Earth on the electron? Recall Earth’s mass 6.0 x 10​24​ kg
and radius 6.4 x 10​6​ m.
c) Based on your results, what can you conclude about these interactions?
4.6 Represent and Reason ​The spheres in the illustration below have equal magnitude charge and are very
light. They are connected to the cart and each other by rods made of an insulating material (also very light).
The cart rests on a smooth table.
a. In this instance, consider​ only electric forces—ignore other types of forces.
Draw a force diagram
for the electric forces
exerted on the left
sphere. Represent the
object with a dot.
Draw a force diagram
for the electric forces
exerted on the center
sphere.
Draw a force diagram
for the electric forces
exerted on the right
sphere.
Draw a force diagram
for the electric forces
exerted on the whole
cart (a system with all
three charged spheres
included).
b. Will the cart accelerate to the left or to the right? Explain your answer.
4.7 Regular Problem​ An object with a charge of 1.0 C and an object with a charge of 2.0 C are held in
place a distance of 100 m from each other. Where should an object with a –1.0 x 10​-3​ C charge be located on
the line between the two positively charged objects so that the net electrical force on the third object is zero?
Note: In real life, objects never hold 1 or 2 C of charge. Why might this be?
4.8 Regular Problem
Three tiny charged spheres lie along the x-axis. The first is located at -0.1 m and has an excess charge of -5
μC. The second tiny sphere is located at 0.3 m and has an excess charge of +2 μC. A third tiny sphere, with
an excess charge of +3 μC, is placed somewhere along the axis.
a. Determine the net force exerted on the third sphere when it is located at the origin.
b. Determine where the third sphere is located when the net force exerted on the sphere is zero.
c. Qualitatively, where could you place the sphere such that the magnitude of the net force is greatest?
4.9 Regular problem​ Two small plastic spheres, which were rubbed with fur, have an excess of negative
charge on them. The two spheres are 20 cm apart from each other; they exert an electrostatic force of 0.5 N
on each other. What is the excess charge on each sphere if the two spheres have the same charge? If one
sphere has 6 times more negative charged particles than the other?
Buggé: Electrostatics 4
4.10 ​Two equal-mass stationary spheres are attached to the end of two strings. The sphere on the left has an
electric charge of +5​Q and the sphere on the right has an electric charge of +​Q. Each string makes an angle
less than 45​o​ with respect to the vertical. Fill in the table that follows.
Draw a force
diagram for the
left sphere.
Draw a force
diagram for the
right sphere.
Decide which
string makes a
bigger angle
with the vertical
or if they make
the same angle.
Apply Newton’s
second law in
component form for
the right sphere.
Based on your analysis, rank order the
forces ​TS on ​Q, F5​Q on​ Q, and ​F E on ​Q,
listing the largest force first.
Horizontal x-axis:
Explain the ranking:
Vertical y-axis:
4.11 ​Two small plastic spheres, each with a mass of 30 grams, are suspended by light strings (L = 15 cm)
from a common point. The spheres are given the same electric charge and the spheres repel each other. Once
the system is in equilibrium, each string makes an angle of 8º with the vertical. Determine the magnitude of
the excess charge on each sphere. (Answer: q = 8.6 x 10​-8​ C, F​string on q1​ = 0.3 N)
4.12 ​Three metal spheres have the following electric charges: q​A​ = + 2.0 x 10​-9​ C, q​B​ = + 2.0 x 10​-9​ C, and
q​c​ = - 4.0 x 10​-9​ C. The spheres are placed at corners of an equilateral triangle whose sides have length x =
1.0 m with C at the top of the triangle. What is the magnitude of the total electric force that spheres A and B
exert on C? (Answer: magnitude 6.2 x 10​-8​ N (times 2), straight down)
4.13 ​A small spherical insulator of mass 8.00 x 10​-2 ​kg and charge +
0.600 μC is hung by a thin wire of negligible mass. A charge of – 0.900
μC is held 0.15 m away from the sphere and directly to the right of it, so
the wire makes an angle θ with the vertical. Find (a) the angle θ and (b)
the tension in the wire. (Answer: 15.4 degrees, 0.813 N)